Theorem 0nnq 10853

Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
0nnq	⊢ ¬ ∅ ∈ Q

Proof of Theorem 0nnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelxp 5665	. 2 ⊢ ¬ ∅ ∈ (N × N)
2		df-nq 10841	. . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
3	2	ssrab3 4041	. . 3 ⊢ Q ⊆ (N × N)
4	3	sseli 3939	. 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N))
5	1, 4	mto 197	1 ⊢ ¬ ∅ ∈ Q

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2109  ∀wral 3044  ∅c0 4292   class class class wbr 5102   × cxp 5629  ‘cfv 6499  2^nd c2nd 7946  Ncnpi 10773   <_N clti 10776   ~_Q ceq 10780  Qcnq 10781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-opab 5165  df-xp 5637  df-nq 10841

This theorem is referenced by:  adderpq  10885  mulerpq  10886  addassnq  10887  mulassnq  10888  distrnq  10890  recmulnq  10893  recclnq  10895  ltanq  10900  ltmnq  10901  ltexnq  10904  nsmallnq  10906  ltbtwnnq  10907  ltrnq  10908  prlem934  10962  ltaddpr  10963  ltexprlem2  10966  ltexprlem3  10967  ltexprlem4  10968  ltexprlem6  10970  ltexprlem7  10971  prlem936  10976  reclem2pr  10977